A328321 Numbers n for which A328311(n) = 1 + A051903(A003415(n)) - A051903(n) is strictly positive.

4, 6, 10, 12, 14, 15, 16, 20, 21, 22, 26, 27, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 48, 50, 51, 52, 54, 55, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 91, 92, 93, 94, 95, 99, 100, 102, 105, 106, 108, 110, 111, 112, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 132, 133

Offset

1,1

Comments

Numbers n for which A051903(A003415(n)) >= A051903(n), i.e., numbers such that taking their arithmetic derivative does not decrease their "degree", A051903, the maximal exponent in prime factorization.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Example

10 = 2*5 has maximal exponent (A051903) 1, and its arithmetic derivative A003415(10) = 2+5 = 7 also has maximal exponent 1, thus 10 is included in this sequence.
15 = 3*5 has maximal exponent 1, and its arithmetic derivative A003415(15) = 3+5 = 8 = 2^3 has maximal exponent 3, thus 15 is included in this sequence.
For 8 = 2^3, its arithmetic derivative A003415(8) = 12 = 2^2 * 3, and as 2 < 3 (highest exponent of 12 is less than that of 8), 8 is NOT included here, and from this we also see that A100716 is not a subsequence of this sequence.

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1, 0, 1+(A051903(A003415(n)) - A051903(n)));
isA328321(n) = (A328311(n)>0);

Crossrefs

Cf. A003415, A051903, A100716, A328302, A328310, A328311.

Cf. A328320 (complement), A051674, A157037, A328304, A328305 (subsequences).

Sequence in context: A137877 A246022 A174240 * A287342 A309177 A163164

Adjacent sequences: A328318 A328319 A328320 * A328322 A328323 A328324

Keyword

nonn

Author

Antti Karttunen, Oct 13 2019

Status

approved